McGraw-Hill/Irwin © 2008 The McGraw-Hill Companies, Inc., All Rights Reserved. Risk and Return: Past and Prologue CHAPTER 5

RATES OF RETURN

5-3 Holding Period Return

5-4 Rates of Return: Single Period Example Ending Price = 24 Beginning Price = 20 Dividend = 1 HPR = ( )/ ( 20) = 25%

5-5 Measuring Investment Returns Over Multiple Periods May need to measure how a fund performed over a preceding five-year period Return measurement is more ambiguous in this case

5-6 Rates of Return: Multiple Period Example Text (Page 128) Data from Table Assets(Beg.) HPR (.20).25 TA (Before Net Flows) Net Flows (0.8) 0.0 End Assets

5-7 Returns Using Arithmetic and Geometric Averaging Arithmetic r a = (r 1 + r 2 + r r n ) / n r a = ( ) / 4 =.10 or 10% =.10 or 10%Geometric r g = {[(1+r 1 ) (1+r 2 ).... (1+r n )]} 1/n - 1 r g = {[(1.1) (1.25) (.8) (1.25)]} 1/4 - 1 = (1.5150) 1/4 -1 =.0829 = 8.29% = (1.5150) 1/4 -1 =.0829 = 8.29%

5-8 Dollar Weighted Returns Internal Rate of Return (IRR) - the discount rate that results in present value of the future cash flows being equal to the investment amount Internal Rate of Return (IRR) - the discount rate that results in present value of the future cash flows being equal to the investment amount Considers changes in investment Initial Investment is an outflow Ending value is considered as an inflow Additional investment is a negative flow Reduced investment is a positive flow

5-9 Dollar Weighted Average Using Text Example (Page 128) Net CFs $ (mil)

5-10 Quoting Conventions APR = annual percentage rate (periods in year) X (rate for period) EAR = effective annual rate ( 1+ rate for period) Periods per yr - 1 Example: monthly return of 1% APR = 1% X 12 = 12% EAR = (1.01) = 12.68%

RISK AND RISK PREMIUMS

5-12 Scenario Analysis and Probability Distributions 1) Mean: most likely value 2) Variance or standard deviation 3) Skewness * If a distribution is approximately normal, the distribution is described by characteristics 1 and 2

5-13 r r Symmetric distribution Normal Distribution s.d.

5-14 r r NegativePositive Skewed Distribution: Large Negative Returns Possible Median

5-15 r rNegativePositive Skewed Distribution: Large Positive Returns Possible Median

5-16 Subjective returns p(s) = probability of a state r(s) = return if a state occurs s = states, s=1,2,3,…,S p(s) = probability of a state r(s) = return if a state occurs s = states, s=1,2,3,…,S Measuring Mean: Scenario or Subjective Returns

5-17 Numerical Example: Subjective or Scenario Distributions StateProb. of Stater in State E(r) = (.1)(-.05) + (.2)(.05)...+ (.1)(.35) E(r) =.15 or 15%

5-18 Measuring Variance or Dispersion of Returns Subjective or Scenario

5-19 Measuring Variance or Dispersion of Returns Using Our Example: Var =[(.1)( ) 2 + (.2)( ) ( ) 2 ] Var= S.D.= [.01199] 0.5 =.1095 or 10.95%

5-20 Risk Premiums and Risk Aversion Degree to which investors are willing to commit funds –Risk aversion If T-Bill denotes the risk-free rate, r f, and variance,, denotes volatility of returns then: The risk premium of a portfolio is:

5-21 Risk Premiums and Risk Aversion To quantify the degree of risk aversion with parameter A: Or:

5-22 The Sharpe (Reward-to-Volatility) Measure

THE HISTORICAL RECORD

5-24 Annual Holding Period Returns From Table 5.3 of Text Geom.Arith.Stan. SeriesMean%Mean%Dev.% World Stk US Lg Stk US Sm Stk Wor Bonds LT Treas T-Bills Inflation

5-25 Annual Holding Period Excess Returns From Table 5.3 of Text Risk Stan. Sharpe SeriesPrem. Dev.% Measure World Stk US Lg Stk US Sm Stk Wor Bonds LT Treas

5-26 Figure 5.1 Frequency Distributions of Holding Period Returns

5-27 Figure 5.2 Rates of Return on Stocks, Bonds and T-Bills

5-28 Figure 5.3 Normal Distribution with Mean of 12% and St Dev of 20%

5-29 Table 5.4 Size-Decile Portfolios

INFLATION AND REAL RATES OF RETURN

5-31 Real vs. Nominal Rates Fisher effect: Approximation nominal rate = real rate + inflation premium R = r + i or r = R - i Example r = 3%, i = 6% R = 9% = 3% + 6% or 3% = 9% - 6% Fisher effect: Exact r = (R - π) / (1 + π) 2.83% = (9%-6%) / (1.06)

5-32 Real vs. Nominal Rates Fisher effect: 2.83% = (9%-6%) / (1.06)

5-33 Figure 5.4 Interest, Inflation and Real Rates of Return

ASSET ALLOCATION ACROSS RISKY AND RISK-FREE PORTFOLIOS

5-35 Possible to split investment funds between safe and risky assets Risk free asset: proxy; T-bills Risky asset: stock (or a portfolio) Allocating Capital

5-36 Allocating Capital Issues –Examine risk/ return tradeoff –Demonstrate how different degrees of risk aversion will affect allocations between risky and risk free assets

5-37 Some Formulas from Statistics Y, X are random variables a, b are parameters (not random) E(aX + bY) = aE(X) + bE(Y) V(aX + bY) = a 2 V(X) + b 2 V(Y) + 2abCov(X,Y) Corr(X,Y)=Cov(X,Y)/{[V(x)V(Y)].5 } = Cov(X,Y)/{SD(x)SD(Y)}

5-38 The Risky Asset: Text Example (Page 143) Total portfolio value = $300,000 Risk-free value = 90,000 Risky (Vanguard and Fidelity) = 210,000 Vanguard (V) = 54% Fidelity (F) = 46%

5-39 The Risky Asset: Text Example (Page 143) Vanguard113,400/300,000 = Fidelity 96,600/300,000 = Portfolio P210,000/300,000 = Risk-Free Assets F 90,000/300,000 = Portfolio C300,000/300,000 = 1.000

5-40 r f = 7%  rf = 0% E(r p ) = 15%  p = 22% y = % in p (1-y) = % in r f Calculating the Expected Return Text Example (Page 145)

5-41 E(r c ) = yE(r p ) + (1 - y)r f r c = complete or combined portfolio For example, y =.75 E(r c ) =.75(.15) +.25(.07) =.13 or 13% Expected Returns for Combinations

5-42 Figure 5.5 Investment Opportunity Set with a Risk-Free Investment

5-43 p p c c = = Since rfrf rfrf y y Variance on the Possible Combined Portfolios = 0, then      

5-44 c c =.75(.22) =.165 or 16.5% If y =.75, then c c = 1(.22) =.22 or 22% If y = 1 c c = 0(.22) =.00 or 0% If y = 0 Combinations Without Leverage      

5-45 Using Leverage with Capital Allocation Line Borrow at the Risk-Free Rate and invest in stock Borrow at the Risk-Free Rate and invest in stock Using 50% Leverage r c = (-.5) (.07) + (1.5) (.15) =.19  c = (1.5) (.22) =.33

5-46 Figure 5.6 Investment Opportunity Set with Differential Borrowing and Lending Rates

5-47 Risk Aversion and Allocation Greater levels of risk aversion lead to larger proportions of the risk free rate Lower levels of risk aversion lead to larger proportions of the portfolio of risky assets Willingness to accept high levels of risk for high levels of returns would result in leveraged combinations

PASSIVE STRATEGIES AND THE CAPITAL MARKET LINE

5-49 Table 5.5 Average Rates of Return, Standard Deviation and Reward to Variability

5-50 Costs and Benefits of Passive Investing Active strategy entails costs Free-rider benefit Involves investment in two passive portfolios –Short-term T-bills –Fund of common stocks that mimics a broad market index